The random Kakutani fixed point theorem in random normed modules
Qiang Tu, Xiaohuan Mu, Tiexin Guo, Guang Yang, Yuanyuan Sun
TL;DR
The paper addresses the existence of fixed points for set-valued mappings on random sequentially compact, $L^{0}$-convex subsets of random normed modules (RN modules). It develops a random Schauder projection framework and leverages a noncompact Schauder fixed point theorem to prove a random Kakutani fixed point theorem, extending the classical Kakutani result to the random setting and generalizing the noncompact Schauder theorem to RN modules. The main result establishes that a $\sigma$-stable $\mathcal{T}_{c}$-upper semicontinuous set-valued map with closed, $L^{0}$-convex values has a fixed point on a random sequentially compact $L^{0}$-convex domain. This provides a foundational fixed point tool in random functional analysis with potential applications to stochastic equilibria and random optimization. The work connects random sequential compactness, random total boundedness, and stable concatenation techniques to extend classical fixed point theory into the RN-module framework, enabling new avenues in stochastic analysis and dynamic games under uncertainty.
Abstract
Based on the recently developed theory of random sequential compactness, we prove the random Kakutani fixed point theorem in random normed modules: if G is a random sequentially compact L0-convex subset of a random normed module, then every -stable Tc-upper semicontinuous mapping F:G to 2G such that F(x) is closed and L0-convex for each x in G, has a fixed point. This is the first fixed point theorem for set-valued mappings in random normed modules, providing a random generalization of the classical Kakutani fixed point theorem as well as a set-valued extension of the noncompact Schauder fixed point theorem established in Math. Ann. 391(3), 3863--3911 (2025).